Computability of analog networks

Theoretical Computer Science

Volumn 371, Issue 1-2, 2007, Pages 115-146

  Tucker, John V ^a   Zucker, Jeffery I ^b  

^a SWANSEA UNIVERSITY   (United Kingdom)

^b MCMASTER UNIVERSITY   (Canada)

Author keywords

Analog computation; Analog computing; Analog networks; Continuous streams; Continuous time

Indexed keywords

COMPUTATION THEORY; COMPUTER NETWORKS; DATA PROCESSING; FUNCTION EVALUATION; MATHEMATICAL MODELS; PARAMETER ESTIMATION; PROBLEM SOLVING;

ANALOG COMPUTATION; ANALOG COMPUTING; ANALOG NETWORKS; CONTINUOUS STREAMS; CONTINUOUS TIME; MODULE FUNCTIONS;

COMPUTABILITY AND DECIDABILITY;

EID: 33846805655     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2006.10.018     Document Type: Article

References (47)

1. Bournez O., Campagnolo M.L., Graça D.S., and Hainry E. The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation. In: Cai J.Y., Cooper S.B., and Li A. (Eds). Theory and Applications of Models of Computation: Third International Conference. TAMC 2006, Beijing, China, 15-20 May 2006, Proceedings. Lecture Notes in Computer Science vol. 3959 (2006), Springer-Verlag 631-643
2. O. Bournez, M.L. Campagnolo, D.S. Graça, E. Hainry, Polynomial differential equations compute all real computable functions, Journal of Complexity (2007) (in press)
3. Beeson M. Foundations of Constructive Mathematics (1985), Springer-Verlag
4. Blanck J., Stoltenberg-Hansen V., and Tucker J.V. Domain representation of streams. In: Möller B., and Tucker J.V. (Eds). Prospects for Hardware Foundations. Lecture Notes in Computer Science vol. 1546 (1998), Springer-Verlag
5. Bush V. The differential analyzer. A new machine for solving differential equations. Journal of the Franklin Institute 212 (1931) 447-488
6. Courant R., and Hilbert D. Methods of Mathematical Physics vol. II (1953), Interscience Translated and revised from the German, 1937
7. Campagnolo M., Moore C., and Costa J.F. An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity 18 (2002) 977-1000
8. Cull P. Dynamics of neural nets. Trends in Biological Cybernetics 1 (1991) 331-349
9. Engelking R. General Topology (1989), Heldermann Verlag
10. Graça D.S., and Costa J.F. Analog computers and recursive functions over the reals. Journal of Complexity 19 (2003) 644-664
11. Graça D.S., Campagnolo M.L., and Buescu J. Robust simulations of Turing machines with analyic maps and flows. In: Cooper S.B., Löwe B., and Torenvliet L. (Eds). New Computational Paradigms: First Conference on Computability in Europe. CiE 2005, Amsterdam, The Netherlands, June 2005: Proceedings. Lecture Notes in Computer Science vol. 3526 (2005), Springer-Verlag 169-179
12. In: Griffor E. (Ed). Handbook of Computability Theory (1999), North Holland
13. Hadamard J. Lectures on Cauchy's Problem in Linear Partial Differential Equations (1952), Dover Translated from the French, 1922
14. Hadamard J. La Théorie des Équations aux Dérivées Partielles (1964), Éditions Scientifiques
15. Hartree D. Calculating Machines and Instruments (1950), Cambridge University Press
16. Haykin S. An Introduction to Analog and Digital Communications (1989), John Wiley & Sons
17. Holst P.A. Svein Rosseland and the Oslo Analyzer. IEEE Annals in the History of Computing 18 (1996) 16-26
18. Hobley K.M., Thompson B.C., and Tucker J.V. Specification and verification of synchronous concurrent algorithms: A case study of a convoluted algorithm. In: Milne G. (Ed). The Fusion of Hardware Design and Verification (Proceedings of IFIP Working Group 10.2 Working Conference) (1988), North Holland 347-374
19. Hyndman D.E. Analog and Hybrid Computing (1970), Pergamon Press
20. Johansson M. Early analog computers in Sweden - with examples from Chalmers University of Technology and the Swedish aerospace industry. IEEE Annals in the History of Computing 18 (1996) 27-33
21. Kleene S.C. Introduction to Metamathematics (1952), North Holland
22. Lipshitz L., and Rubel L.A. A differentially algebraic replacement theorem. Proceedings of the American Mathematical Society 99 2 (1987) 367-372
23. Mycka J., and Costa J.F. Real recursive functions and their hierarchy. Journal of Complexity 20 (2004) 835-857
24. Moore C. Recursion theory on the reals and continuous time computation. Theoretical Computer Science 162 (1996) 23-44
25. Myrvold W.C. Computability in quantum mechanics. In: DePauli-Schimanovich W., Köhler E., and Stadler F. (Eds). The Foundational Debate: Complexity and Constructivity in Mathematics and Physics (1995), Kluwer 33-46
26. Oppenheim A.V., and Willsky A.S. Signals and Systems. 2nd edition (1997), Prentice Hall
27. Pour-El M.B. Abstract computability and its relation to the general-purpose analog computer. Journal of the American Mathematical Society 199 (1974) 1-28
28. Pour-El M.B., and Richards J.I. Computability in Analysis and Physics (1989), Springer-Verlag
29. Puchta S. On the role of mathematics and mathematical knowledge in the invention of Vannevar Bush's early analog computers. IEEE Annals in the History of Computing 18 (1996) 49-56
30. Rabinovich A. Automata over continuous time. Theoretical Computer Science 300 (2003) 331-363
31. Royden H.L. Real Analysis (1963), Macmillan
32. Rabinovich A., and Trakhtenbrot B.A. From finite automata toward hybrid systems. In: Brim L., Gruska J., and Zlatuska J. (Eds). Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science. MFCS'98, Brno, Czech Republic, August 1998. Lecture Notes in Computer Science vol. 1450 (1998), Springer-Verlag 411-422
33. Rudin W. Principles of Mathematical Analysis. 3rd edition (1976), McGraw-Hill
34. Shannon C. Mathematical theory of the differential analyser. Journal of Mathematics and Physics 20 (1941) 337-354
35. Stoltenberg-Hansen V., and Tucker J.V. Concrete models of computation for topological algebras. Theoretical Computer Science 219 (1999) 347-378
36. Small J.S. General-purpose electronic analog computing: 1945-1965. IEEE Annals in the History of Computing 18 (1996) 8-18
37. Sontag E.D. Mathematical Control Theory: Deterministic Finite Dimensional Systems (1990), Springer-Verlag
38. Trakhtenbrot B.A. Finite automata and the logic of one-place predicates. AMS Translations 59 (1966) 23-55 Translated from the Russian, 1961
39. Thompson W., and Tait P.G. Treatise on Natural Philosophy. 2nd edition, Part I (1880), Cambridge University Press
40. B.C. Thompson, J.V. Tucker, Algebraic specification of synchronous concurrent algorithms and architectures (Revised), Research Report 9-91, Department of Computer Science, University College of Swansea, 1991
41. Tucker J.V., and Zucker J.I. Computable functions on stream algebras. In: Schwichtenberg H. (Ed). Proof and Computation: NATO Advanced Study Institute International Summer School at Marktoberdorf, 1993 (1994), Springer-Verlag 341-382
42. Tucker J.V., and Zucker J.I. Abstract versus concrete computation on metric partial algebras. ACM Transactions on Computational Logic 5 (2004) 611-668
43. Tucker J.V., and Zucker J.I. Computable total functions, algebraic specifications and dynamical systems. Journal of Logic and Algebraic Programming 62 (2005) 71-108
44. Tucker J.V., and Zucker J.I. Abstract versus concrete computability: The case of countable algebras. In: Stoltenberg-Hansen V., and Väänänen J. (Eds). Logic Colloquium '03, Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic. Helsinki, Finland, 14-20 August 2003. Lecture Notes in Logic vol. 24 (2006), Association for Symbolic Logic 377-408
45. J.V. Tucker, J.I. Zucker, Computation on algebras of continuous functions, 2007 (in preparation)
46. Weihrauch K. Computable Analysis: An Introduction (2000), Springer-Verlag
47. Wigner E. The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure Mathematics 13 (1960) 1-14